jmc

There is $1$ integer whose digits sum to $1$: $10$. There are $4$ integers whose digits sum to $4$: $13,22,31,\text{and }40$. There are $9$ integers whose digits sum to $9$: $18,27,36,45,54,63,72,81,\text{and }90$. There are $3$ integers whose digits sum to $16$: $79,88,\text{and }97$. Two digits cannot sum to $25$ or any greater square since the greatest sum of digits of a two-digit number is $9+9=18$. Thus, the answer is $1+4+9+3=\boxed{17}$.